Abstract

We define and study precision and consistency criteria for spectral phase interferometry for direct electric-field reconstruction (SPIDER), an interferometric technique for the characterization of ultrashort optical pulses. Precision quantifies the similarity of different estimates of the electric field reconstructed from a single experimental data set. Consistency is a measure of the match between the reconstructed field and the data. These powerful experimental criteria allow one to monitor the fidelity of the reconstruction of the electric field from the experimental data by tracking systematic errors and random noise. Because of such imperfections, such criteria are necessary even for pulse characterization methods based on a direct algebraic inversion. In the case of SPIDER, the redundancy of data in each interferogram and the principle of shearing interferometry allow simple calculation of the precision and consistency measures.

© 2002 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  6. M. Munroe, D. H. Christensen, and R. Trebino, “Error bars in intensity and phase measurements of ultrashort laser pulses,” Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper CThW2.
  7. R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
    [Crossref]
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    [Crossref]
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    [Crossref]
  10. R. G. M. P. Koumans and A. Yariv, “Time-resolved optical gating based on dispersive propagation: a new method to characterize optical pulses,” IEEE J. Quantum Electron. 36, 137–144 (2000).
    [Crossref]
  11. D. J. Kane, “Recent progress toward real-time measurement of ultrashort laser pulses,” IEEE J. Quantum Electron. 35, 421–431 (1999).
    [Crossref]
  12. D. J. Kane, F. G. Omenetto, and A. J. Taylor, “Convergence test for inversion of frequency-resolved optical gating spectrograms,” Opt. Lett. 25, 1216–1218 (2000).
    [Crossref]
  13. J. Paye, “How to measure the amplitude and phase of an ultrashort light pulse with an autocorrelator and a spectrometer,” IEEE J. Quantum Electron. 30, 2693–2697 (1994).
    [Crossref]
  14. C. Iaconis and I. A. Walmsley, “Spectral phase interferom-etry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
    [Crossref]
  15. L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, and I. A. Walmsley, “Characterization of sub-6-fs optical pulses with spectral phase interferom-etry for direct electric-field reconstruction,” Opt. Lett. 24, 1314–1316 (1999).
    [Crossref]
  16. C. Dorrer, B. de Beauvoir, C. Le Blanc, S. Ranc, J. P. Rousseau, P. Rousseau, and J. P. Chambaret, “Single-shot real-time characterization of chirped-pulse amplification systems by spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24, 1644–1646 (1999).
    [Crossref]

2002 (1)

2000 (3)

M. E. Anderson, L. E. E. de Araujo, E. M. Kosik, and I. A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using spectral phase interferometry for direct electric-field reconstruction,” Appl. Phys. B 70, S85–S93 (2000).
[Crossref]

R. G. M. P. Koumans and A. Yariv, “Time-resolved optical gating based on dispersive propagation: a new method to characterize optical pulses,” IEEE J. Quantum Electron. 36, 137–144 (2000).
[Crossref]

D. J. Kane, F. G. Omenetto, and A. J. Taylor, “Convergence test for inversion of frequency-resolved optical gating spectrograms,” Opt. Lett. 25, 1216–1218 (2000).
[Crossref]

1999 (3)

1998 (2)

C. Iaconis and I. A. Walmsley, “Spectral phase interferom-etry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[Crossref]

H. Lange, M. Franco, J-F. Ricpoche, B. Prade, P. Rousseau, and A. Mysyrowicz, “Reconstruction of the time profile of femtosecond laser pulses through cross-phase modulation,” IEEE J. Sel. Top. Quantum Electron. 4, 295–300 (1998).
[Crossref]

1997 (3)

1995 (2)

1994 (1)

J. Paye, “How to measure the amplitude and phase of an ultrashort light pulse with an autocorrelator and a spectrometer,” IEEE J. Quantum Electron. 30, 2693–2697 (1994).
[Crossref]

Anderson, M. E.

M. E. Anderson, L. E. E. de Araujo, E. M. Kosik, and I. A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using spectral phase interferometry for direct electric-field reconstruction,” Appl. Phys. B 70, S85–S93 (2000).
[Crossref]

Chambaret, J. P.

Christensen, D. H.

M. Munroe, D. H. Christensen, and R. Trebino, “Error bars in intensity and phase measurements of ultrashort laser pulses,” Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper CThW2.

Clement, T. S.

de Araujo, L. E. E.

M. E. Anderson, L. E. E. de Araujo, E. M. Kosik, and I. A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using spectral phase interferometry for direct electric-field reconstruction,” Appl. Phys. B 70, S85–S93 (2000).
[Crossref]

de Beauvoir, B.

Delong, K. W.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

D. N. Fittinghoff, K. W. Delong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort optical pulses,” J. Opt. Soc. Am. B 12, 1955–1967 (1995).
[Crossref]

Dorrer, C.

Fittinghoff, D. N.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

D. N. Fittinghoff, K. W. Delong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort optical pulses,” J. Opt. Soc. Am. B 12, 1955–1967 (1995).
[Crossref]

Franco, M.

H. Lange, M. Franco, J-F. Ricpoche, B. Prade, P. Rousseau, and A. Mysyrowicz, “Reconstruction of the time profile of femtosecond laser pulses through cross-phase modulation,” IEEE J. Sel. Top. Quantum Electron. 4, 295–300 (1998).
[Crossref]

Gallmann, L.

Iaconis, C.

Kane, D. J.

D. J. Kane, F. G. Omenetto, and A. J. Taylor, “Convergence test for inversion of frequency-resolved optical gating spectrograms,” Opt. Lett. 25, 1216–1218 (2000).
[Crossref]

D. J. Kane, “Recent progress toward real-time measurement of ultrashort laser pulses,” IEEE J. Quantum Electron. 35, 421–431 (1999).
[Crossref]

D. J. Kane, G. Rodriguez, A. J. Taylor, and T. S. Clement, “Simultaneous measurement of two ultrashort laser pulses from a single spectrogram in a single shot,” J. Opt. Soc. Am. B 14, 935–943 (1997).
[Crossref]

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Keller, U.

Kosik, E. M.

M. E. Anderson, L. E. E. de Araujo, E. M. Kosik, and I. A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using spectral phase interferometry for direct electric-field reconstruction,” Appl. Phys. B 70, S85–S93 (2000).
[Crossref]

Koumans, R. G. M. P.

R. G. M. P. Koumans and A. Yariv, “Time-resolved optical gating based on dispersive propagation: a new method to characterize optical pulses,” IEEE J. Quantum Electron. 36, 137–144 (2000).
[Crossref]

Krumbugel, M. A.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Ladera, C. L.

Lange, H.

H. Lange, M. Franco, J-F. Ricpoche, B. Prade, P. Rousseau, and A. Mysyrowicz, “Reconstruction of the time profile of femtosecond laser pulses through cross-phase modulation,” IEEE J. Sel. Top. Quantum Electron. 4, 295–300 (1998).
[Crossref]

Le Blanc, C.

Matuschek, N.

Munroe, M.

M. Munroe, D. H. Christensen, and R. Trebino, “Error bars in intensity and phase measurements of ultrashort laser pulses,” Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper CThW2.

Mysyrowicz, A.

H. Lange, M. Franco, J-F. Ricpoche, B. Prade, P. Rousseau, and A. Mysyrowicz, “Reconstruction of the time profile of femtosecond laser pulses through cross-phase modulation,” IEEE J. Sel. Top. Quantum Electron. 4, 295–300 (1998).
[Crossref]

Omenetto, F. G.

Paye, J.

J. Paye, “How to measure the amplitude and phase of an ultrashort light pulse with an autocorrelator and a spectrometer,” IEEE J. Quantum Electron. 30, 2693–2697 (1994).
[Crossref]

Prade, B.

H. Lange, M. Franco, J-F. Ricpoche, B. Prade, P. Rousseau, and A. Mysyrowicz, “Reconstruction of the time profile of femtosecond laser pulses through cross-phase modulation,” IEEE J. Sel. Top. Quantum Electron. 4, 295–300 (1998).
[Crossref]

Ranc, S.

Richman, B. A.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Ricpoche, J-F.

H. Lange, M. Franco, J-F. Ricpoche, B. Prade, P. Rousseau, and A. Mysyrowicz, “Reconstruction of the time profile of femtosecond laser pulses through cross-phase modulation,” IEEE J. Sel. Top. Quantum Electron. 4, 295–300 (1998).
[Crossref]

Rodriguez, G.

Rousseau, J. P.

Rousseau, P.

C. Dorrer, B. de Beauvoir, C. Le Blanc, S. Ranc, J. P. Rousseau, P. Rousseau, and J. P. Chambaret, “Single-shot real-time characterization of chirped-pulse amplification systems by spectral phase interferometry for direct electric-field reconstruction,” Opt. Lett. 24, 1644–1646 (1999).
[Crossref]

H. Lange, M. Franco, J-F. Ricpoche, B. Prade, P. Rousseau, and A. Mysyrowicz, “Reconstruction of the time profile of femtosecond laser pulses through cross-phase modulation,” IEEE J. Sel. Top. Quantum Electron. 4, 295–300 (1998).
[Crossref]

Steinmeyer, G.

Sutter, D. H.

Sweetser, J. N.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Taylor, A. J.

Trebino, R.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

D. N. Fittinghoff, K. W. Delong, R. Trebino, and C. L. Ladera, “Noise sensitivity in frequency-resolved optical-gating measurements of ultrashort optical pulses,” J. Opt. Soc. Am. B 12, 1955–1967 (1995).
[Crossref]

M. Munroe, D. H. Christensen, and R. Trebino, “Error bars in intensity and phase measurements of ultrashort laser pulses,” Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper CThW2.

Walmsley, I.

Walmsley, I. A.

Wong, V.

Yariv, A.

R. G. M. P. Koumans and A. Yariv, “Time-resolved optical gating based on dispersive propagation: a new method to characterize optical pulses,” IEEE J. Quantum Electron. 36, 137–144 (2000).
[Crossref]

Appl. Phys. B (1)

M. E. Anderson, L. E. E. de Araujo, E. M. Kosik, and I. A. Walmsley, “The effects of noise on ultrashort-optical-pulse measurement using spectral phase interferometry for direct electric-field reconstruction,” Appl. Phys. B 70, S85–S93 (2000).
[Crossref]

IEEE J. Quantum Electron. (3)

J. Paye, “How to measure the amplitude and phase of an ultrashort light pulse with an autocorrelator and a spectrometer,” IEEE J. Quantum Electron. 30, 2693–2697 (1994).
[Crossref]

R. G. M. P. Koumans and A. Yariv, “Time-resolved optical gating based on dispersive propagation: a new method to characterize optical pulses,” IEEE J. Quantum Electron. 36, 137–144 (2000).
[Crossref]

D. J. Kane, “Recent progress toward real-time measurement of ultrashort laser pulses,” IEEE J. Quantum Electron. 35, 421–431 (1999).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

H. Lange, M. Franco, J-F. Ricpoche, B. Prade, P. Rousseau, and A. Mysyrowicz, “Reconstruction of the time profile of femtosecond laser pulses through cross-phase modulation,” IEEE J. Sel. Top. Quantum Electron. 4, 295–300 (1998).
[Crossref]

J. Opt. Soc. Am. B (5)

Opt. Lett. (4)

Rev. Sci. Instrum. (1)

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Other (1)

M. Munroe, D. H. Christensen, and R. Trebino, “Error bars in intensity and phase measurements of ultrashort laser pulses,” Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper CThW2.

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Figures (10)

Fig. 1
Fig. 1

(a) Spectrum (solid curve) and spectral phases (dashed curves) used in the simulations. (b) Temporal intensity for the flat phase, the parabolic phase, the cubic phase, and the phase jump, from top to bottom; in the time domain, the plotted window is the Nyquist window corresponding to the SPIDER shear.

Fig. 2
Fig. 2

RMS precision error versus noise fraction for the concatenation technique in the case of the phase jump (dashed curve), the flat phase for a broad temporal filter (upper solid curve), and the flat phase for a narrow temporal filter (lower solid curve).

Fig. 3
Fig. 3

Definition of the precision error for the integration algorithms. The initial field E, whose intensity is plotted, is defined on Σ. The average field E is obtained by truncating this field to ΣM, i.e., by setting to zero the field in the gray shaded portions of the plot correpsonding to K. The precision error on the measurement is simply the square root of the energy of E located in these portions.

Fig. 4
Fig. 4

RMS precision error versus noise fraction for the integration technique I1 in the case of the phase jump (dashed curve), the flat phase for a broad temporal filter (upper solid curve), and the flat phase for a narrow temporal filter (lower solid curve).

Fig. 5
Fig. 5

Phase consistency measure in the case of the flat phase (three lower curves), the parabolic phase (three middle curves) and the cubic phase (three upper curves). In each case, the dashed curve corresponds to the trapezoidal rule of integration I1, the ditted curve to the Simspon’s rule of integration I2, and the solid curve to the concatenation technique.

Fig. 6
Fig. 6

Phase consistency measure when reconstructing the phase jump with the low-order integration technique I1 (dashed curve), the high-order integration technique I2 (ditted curve), and the concatenation technique (solid curve).

Fig. 7
Fig. 7

Phase consistency measure for the reconstruction of the flat phase with the low-order integration technique at noise levels of 10-4, 10-3, 10-2, and 10-1 (from bottom to top).

Fig. 8
Fig. 8

Phase consistency measure for the reconstruction of the phase jump with the low-order integration technique at noise levels of 10-4, 10-3, 10-2, and 10-1 (from bottom to top).

Fig. 9
Fig. 9

Integral phase consistency measure versus noise for the reconstruction of the flat phase jump by the integration technique I1 (middle curve), the integration technique I2 (bottom curve), and the concatenation technique (top curve).

Fig. 10
Fig. 10

Integral phase consistency measure versus noise fraction for the reconstruction of the sharp phase jump by the integration technique I1 (middle curve), the integration technique I2 (lower curve), and the concatenation technique (top curve).

Equations (13)

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P=1M mE-Em21/2=1M mEm2-E21/2,
P=(1-E2)1/2.
E(t)=δωnE˜(nδω)exp(-inδωt)inΣ=[-π/δω, π/δω],
E(t)=0elsewhere.
E(t)=δωk=0M-1nE˜[(k+nM)δω]×exp[-i(k+nM)δωt],
E(t)=1M k exp(-ikδωt)MδωnE˜[(k+nM)δω]×exp(-inMδωt).
E(t)=E˜(nδω)exp(-inδωt)inΣM=-πMδω, πMδω,
E(t)=0elsewhere.
P=1M Em2-E21/2=(E2-E2)1/2=K|E(t)|21/2.
Γ(ω)=φ(ω+Ω)-φ(ω).
1Ω ω0ω0+Ωφ(ω)dω=1Ω -ω0Γ(ω)dω.
θ(ω)=1Ω ω0ω0+ΩφR(ω)dω-1Ω -ω0Γ(ω)dω.
Θ=-+|E˜(ω)(1-exp(iθω)|2dω1/2.

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